1. Regression and Median-Fit Lines (4-6)
Objective: Write equations of best-fit lines using linear regression. Write equations of median-fit lines.

2. Best-Fit Lines
You have learned how to find and write equations for lines of fit by hand.
Many calculators use complex algorithms that find a more precise line of fit called the best-fit line.
One algorithm is called linear regression.
Your calculator may also compute a number called the correlation coefficient.
This number will tell you if your correlation is positive or negative and how closely the equation is modeling the data.
The closer the correlation coefficient is to 1 or -1, the more closely the equation models the data.

3. Example 1
The table shows Ariana’s hourly earnings for the years Use a graphing calculator to write an equation for the best-fit line for the data. Name the correlation coefficient. Round to the nearest ten-thousandth. Let x be the number of years since 2000.
Year
Cost
2001
$10
2002
$10.50
2003
$11
2004
$13
2005
$15
2006
$15.75
2007
$16.50

4. Example 1
Before you begin, you need to make sure the Diagnostic setting on your calculator is on.
You can find this setting under the CATALOG menu by pressing 2nd 0 on your calculator.
Alpha is set when you are in this menu. Press D by using the x-1 button and then scroll down until the arrow on the left is pointing to DiagnosticOn.
Press ENTER twice.

5. Example 1
Enter the data in the table into your calculator by pressing STAT.
Choose 1: Edit.
If there is data in the calculator already, you will need to clear it out. Scroll up until L1 is highlighted and press CLEAR and the ENTER. Repeat this with L2.
Enter the independent variable (x) under L1 and the dependent variable (y) under L2.
Year
Cost
2001
$10
2002
$10.50
2003
$11
2004
$13
2005
$15
2006
$15.75
2007
$16.50

6. Example 1
You may then calculate the regression line by pressing STAT again.
Right arrow over so that the heading CALC is highlighted.
Scroll down to 4: LinReg(ax+b) and press ENTER twice.
The “a” value represents the slope.
The “b” value represents the y-intercept.
The “r” value represents the correlation coefficient.
Use “a” and “b” to write the equation for the best-fit line.
y = 1.21x
r =

7. Check Your Progress Choose the best answer for the following.
The table shows the average body temperature in degrees Celsius of nine insects at a given temperature. Use a graphing calculator to write the equation for the best fit line for that data. Name the correlation coefficient.
y = 0.85x ;
y = 0.95x ;
y = 1.53x ;
y= 1.95x ;
Temperature (˚C)
Air
25.7
30.4
28.7
31.2
31.5
26.2
30.1
18.2
Body
27.0
28.9
31.0
25.6
28.4
31.7
18.7

8. Interpolation and Extrapolation
We can use points on the best-fit line to estimate values that are not in the data.
When we estimate values that are between known values, this is called linear interpolation.
When we estimate a number outside of the range of the data, it is called linear extrapolation.

9. Example 2
The table below shows the points earned by the top ten bowlers in a tournament. Estimate how many points the 15th-ranked bowler earned.
y = -7.9x
y = -7.9(15)
y =
y = 83
Rank 1 2 3 4 5 6 7 8 9 10
Score
210
197
164
158
151
147
144
142
134
132

10. Median-Fit Lines
A second type of fit line that can be found using a graphing calculator is a median-fit line.
The equation of a median-fit line is calculated using the medians of the coordinates of the data points.

11. Example 3
Find the equation of a median-fit line for the data on the bowling tournament in Example 2. Then predict the score of the 20th ranked bowler.
Enter the data into the calculator the same way using the STAT button.
After the data has been entered, go back to STAT and right arrow over so that CALC is highlighted.
Choose 3: Med-Med.
Once again, the “a” value is the slope and the “b” value is the y-intercept.
y = -9x
y = -9(20)
y =
y = 30

12. Air Taxi to Kelley’s Island
Check Your Progress
Choose the best answer for the following.
An air taxi keeps track of how many passengers it carries to various islands. The table shows the number of passengers who have traveled to Kelley’s Island in previous years. Use a regression line to determine how many passengers should the airline expect to go to Kelley’s Island in 2015?
1186 passengers
1702 passengers
1890 passengers
2186 passengers
Air Taxi to Kelley’s Island
Year
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
Number of Passengers
1020
1115
1247
1657
1324
1568
1987
1562
1468
1693
y = 68.7x
y = 68.7(15)
y =

13. Check Your Progress Choose the best answer for the following.
Use the data from the table and a median-fit line to estimate the number of passengers the airline will have in 2015.
1100 passengers
1700 passengers
1900 passengers
2100 passengers
y = 63.9x
y = 63.9(15)
y =